1/2x-3=4.5x+9

Solution for 1/2x-3=4.5x+9 equation:

1/2x-3=4.5x+9

We move all terms to the left:

1/2x-3-(4.5x+9)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-4.5x-9-3=0

We multiply all the terms by the denominator


-(4.5x)*2x-9*2x-3*2x+1=0

We add all the numbers together, and all the variables

-(+4.5x)*2x-9*2x-3*2x+1=0

We multiply parentheses

-8x^2-9*2x-3*2x+1=0

Wy multiply elements

-8x^2-18x-6x+1=0

We add all the numbers together, and all the variables

-8x^2-24x+1=0

a = -8; b = -24; c = +1;
Δ = b2-4ac
Δ = -242-4·(-8)·1
Δ = 608
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{608}=\sqrt{16*38}=\sqrt{16}*\sqrt{38}=4\sqrt{38}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-4\sqrt{38}}{2*-8}=\frac{24-4\sqrt{38}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+4\sqrt{38}}{2*-8}=\frac{24+4\sqrt{38}}{-16}
$